MinT - Best Known (127, 127+73, s)-Nets in Base 3

Best Known (127, 127+73, s)-Nets in Base 3

(127, 127+73, 156)-Net over F₃ — Constructive and digital

Digital (127, 200, 156)-net over F₃, using

10 times m-reduction [i] based on digital (127, 210, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 105, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(127, 127+73, 243)-Net over F₃ — Digital

Digital (127, 200, 243)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+73, 3062)-Net in Base 3 — Upper bound on s

There is no (127, 200, 3063)-net in base 3, because

1 times m-reduction [i] would yield (127, 199, 3063)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 88541 500943 818473 352514 758480 410537 861128 134469 328187 809053 264924 551162 195503 880624 086956 612153 > 3¹⁹⁹ [i]